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I know from two sources that it is (or at least was) unknown whether there are infinitely many geometrically distinct closed geodesics for every Riemannian metric on $S^3$, the 3-sphere (Weinberger, Computers, Rigidity, and Moduli, 1995, p.101; Berger, A Panoramic View of Riemannian Geometry, 2003, p.461). And from the same two sources, that it is known that there is at least one closed geodesic on any compact Riemannian manifold (Lusternick and Fet). My question is whether or not it is known that there are at least two distinct closed geodesics on $S^3$?

On $S^2$ it is known there are at least three simple closed geodesics (Lusternick and Schnirelmann), and infinitely many periodic geodesics (Bangert, Franks, Hingston). It might help an idea I'm considering if it were known there is more than one closed geodesic on $S^3$. Thanks for pointers!

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  • $\begingroup$ This question and your last one had no pictures -- is something wrong? $\endgroup$
    – Allen Knutson
    Commented Sep 24, 2011 at 1:19
  • $\begingroup$ Ha! Yes, I am slipping! I should have a picture even for a reference request! :-) $\endgroup$
    – Joseph O'Rourke
    Commented Sep 24, 2011 at 1:46

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I think the recent work by Huagui Duan and Yiming Long

http://arxiv.org/abs/1008.1458

showed the existence of at least two closed geodesics.

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  • $\begingroup$ "The index quasi-periodicity and multiplicity of closed geodesics." Abstract: "In this paper, we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible Finsler (including Riemannian) manifold of dimension not less than 2." Wonderful!! Thanks so much! $\endgroup$
    – Joseph O'Rourke
    Commented Sep 17, 2011 at 11:23

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